Mathematics one exam outline

Mathematics one

Examination subjects: advanced mathematics, linear algebra, probability theory and mathematical statistics.

Advanced mathematics

I. Function, Limit and Continuity

Examination content

The concept of function and its boundedness, monotonicity, periodicity and parity, the properties of compound function, inverse function, piecewise function and implicit function, and the application of establishing function relationship to graphic elementary function.

Definitions and properties of sequence limit and function limit, left limit and right limit of function, concepts and relationships between infinitesimal and infinitesimal, properties of infinitesimal and four operational limits of infinitesimal comparison limit, two important limits: monotone bounded criterion and pinch criterion;

Concept of Function Continuity Types of Discontinuous Points of Functions Continuity of Elementary Functions Properties of Continuous Functions on Closed Interval

Examination requirements

1. Understand the concept of function, master the expression of function, and establish the function relationship in simple application problems.

2. Understand the boundedness, monotonicity, periodicity and parity of functions.

3. Understand the concepts of compound function and piecewise function, inverse function and implicit function.

4. Grasp the nature and graphics of basic elementary functions and understand the concept of elementary functions.

5. Understand the concept of limit, the concepts of left limit and right limit of function, and the relationship between the existence of function limit and left and right limit.

6. Master the nature of limit and four algorithms.

7. Master two criteria for the existence of limit, and use them to find the limit, and master the method of using two important limits to find the limit.

8. Understand the concepts of infinitesimal and infinity, master the comparison method of infinitesimal, and find the limit with equivalent infinitesimal.

9. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity points.

10. Understand the properties of continuous function and continuity of elementary function, understand the properties of continuous function on closed interval (boundedness, maximum theorem, mean value theorem), and apply these properties.

Second, the differential calculus of unary function

Examination content.

The relationship between the geometric meaning of derivative and differential concept and the derivability and continuity of physical meaning function; Four operations of tangent and normal derivative and differential of plane curve derivative; Differential method of compound function, inverse function, implicit function and function determined by parameter equation; Invariant differential mean value theorem of first-order differential form of higher-order derivative; Lobida's law function; Monotonicity; Discrimination of extreme value function; Graph concavity, inflection point and asymptote function; Graph description function; Maximum value and minimum value; Concept curvature radius of arc differential curvature.

Examination requirements

1. Understand the concepts of derivative and differential, understand the relationship between derivative and differential, understand the geometric meaning of derivative, find the tangent equation and normal equation of plane curve, understand the physical meaning of derivative, describe some physical quantities with derivative, and understand the relationship between function derivability and continuity.

2. Master the four algorithms of derivative and the derivative rule of compound function, and master the derivative formula of basic elementary function. Knowing the four algorithms of differential and the invariance of first-order differential form, we can find the differential of function.

3. Understand the concept of higher-order derivative and find the n-order derivative of simple function.

4. We can find the derivative of piecewise function, implicit function, function determined by parametric equation and inverse function.

5. Understand and apply Rolle theorem, Lagrange mean value theorem, Taylor theorem and Cauchy mean value theorem.

6. Master the method of finding the limit of infinitive with L'H?pital's law.

7. Understand the concept of extreme value of function, master the method of judging monotonicity of function and finding extreme value of function with derivative, master the method of finding maximum and minimum value of function and its simple application.

8. We can judge the concavity and convexity of the function graph by derivative, find the inflection point and horizontal, vertical and oblique asymptotes of the function graph, and describe the function graph.

9. Understand the concepts of curvature and radius of curvature, and calculate curvature and radius of curvature.

3. Integral calculus of unary function

Examination content

The concept of original function and indefinite integral The basic properties of indefinite integral The concept of basic integral formula and the basic properties of the mean value theorem of definite integral The upper limit of integral and the function of its derivative Newton-Leibniz formula The substitution integral method of indefinite integral and definite integral and the application of partial integral definite integral The definite integral expresses and calculates the centroid rational function, the rational formula of trigonometric function and the almost definite integral of simple and unreasonable integral generalized integral.

Examination requirements

1. Understand the concepts of original function and indefinite integral and definite integral.

2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral and the mean value theorem of definite integral, and master the integration methods of method of substitution and integration by parts.

3. Can find the integral of rational function, rational formula of trigonometric function and simple unreasonable function.

4. Understand the function of the upper limit of integral, find its derivative and master Newton-Leibniz formula.

5. Knowing the concept of generalized integral, we can calculate generalized integral.

6. Grasp the representation and calculation of some geometric physical quantities (the area of a plane figure, the arc length of a plane curve, the volume and lateral area of a rotating body, and the volume, work, gravity and pressure of a known solid with the area of a parallel section) and the average value of functions through definite integral.

4. Vector Algebra and Spatial Analytic Geometry

Examination content

The quantitative product of the linear operation vector of the concept vector of the vector and the mixed product of the cross product vector are the conditions that the two vectors are vertically parallel. Coordinate expression of included angle vector of two vectors and its operation unit vector direction number and direction cosine surface equation and space curve equation are plane equation, straight line equation plane to plane, plane to straight line, straight line to straight line parallel, and the distance and straight line between a vertical condition point and plane; Equation of surface of revolution with spherical generatrix parallel to coordinate axis and cylindrical rotation axis as coordinate axis; Commonly used quadratic equation and parametric equation of its graphic space curve and projection curve equation of general equation space curve on coordinate plane.

Examination requirements

1. Understand the spatial rectangular coordinate system and the concept and representation of vectors.

2. Master the operation of vectors (linear operation, quantitative product, cross product, mixed product) and understand the conditions for two vectors to be vertical and parallel.

3. Understand the coordinate expressions of unit vector, direction number, direction cosine and vector, and master the method of vector operation with coordinate expressions.

4. Principal plane equation and straight line equation and their solutions.

5. Will find the angles between planes, between planes and straight lines, and between straight lines, and will use mutual (parallel, vertical, intersecting, etc.). ) plane and straight line to solve related problems.

6. You can find the distance from a point to a straight line and the distance from a point to a plane.

7. Understand the concepts of surface equation and space curve equation.

8. Understand the equations and graphs of quadric surface in common use, and find the cylindrical equations of the rotating surface with the coordinate axis as the rotation axis and the generatrix parallel to the coordinate axis.

9. Understand the parametric equation and general equation of space curve. Understand the projection of space curve on the coordinate plane and find its equation.

Verb (abbreviation of verb) Differential calculus of multivariate functions

Examination content

Concept of multivariate function, geometric meaning of bivariate function, concept of limit and continuity of bivariate function, properties of multivariate continuous function in bounded closed region, necessary and sufficient conditions for existence of partial derivative and total differential of multivariate function, derivation method of implicit function, second-order partial derivative, directional derivative of gradient space curve and second-order Taylor formula of tangent plane and normal plane of multivariate function, maximum and conditional extreme value of multivariate function and its simple application.

Examination requirements

1. Understand the concept of multivariate function and the geometric meaning of bivariate function.

2. Understand the concepts of limit and continuity of binary functions and the properties of continuous functions in bounded closed regions.

3. Understand the concepts of partial derivative and total differential of multivariate function, you will find total differential, understand the necessary and sufficient conditions for the existence of total differential, and understand the invariance of total differential form.

4. Understand the concepts of directional derivative and gradient, and master their calculation methods.

5. Master the solution of the first and second partial derivatives of multivariate composite functions.

6. Knowing the existence theorem of implicit function, we can find the partial derivative of multivariate implicit function.

7. Understand the concepts of tangent and normal plane of space curve and tangent and normal plane of surface, and work out their equations.

8. Understand the second-order Taylor formula of binary function.

9. Understand the concepts of multivariate function extremum and conditional extremum, master the necessary conditions of multivariate function extremum, understand the sufficient conditions of bivariate function extremum, find bivariate function extremum, use Lagrange multiplier method to find conditional extremum, find the maximum and minimum of simple multivariate function, and solve some simple application problems.

Six, multivariate function integral calculus

Examination content

The concepts, properties, calculation and application of double integral and triple integral; The concept, properties and calculation of two kinds of curve integrals: the condition that plane curve integrals have nothing to do with paths; The concepts and properties of two kinds of surface integrals and the calculation of the relationship between the two kinds of surface integrals are known. The concepts of divergence and curl of Gauss formula Stokes formula and its application in calculating curve integral and surface integral.

Examination requirements

1. Understand the concept, properties and mean value theorem of double integral.

2. Master the calculation method of double integrals (rectangular coordinates and polar coordinates), and be able to calculate triple integrals (rectangular coordinates, cylindrical coordinates and spherical coordinates).

3. Understand the concepts, properties and relationships of two kinds of curve integrals.

4. Master the calculation methods of two kinds of curve integrals.

5. Master Green's formula and use the conditions of plane curve integral and path elements to find the original function of total differential.

6. Understand the concepts, properties and relations of two kinds of surface integrals, master the calculation methods of two kinds of surface integrals, and use Gaussian formula and Stokes formula to calculate surface integrals and curve integrals.

7. The concepts of dissolution and rotation are introduced and calculated.

8. We can use multiple integrals, curve integrals and surface integrals to find some geometric physical quantities (area, volume, surface area, arc length, mass, center of gravity, moment of inertia, gravity, work and flow of plane figures, etc.). ).

Seven, infinite series

Examination content

The basic properties and necessary conditions of the convergence of the concept series of the convergence of geometric series and P series of constant series and its convergence criteria of positive series, the absolute convergence and conditional convergence of staggered series and Leibniz theorem, the concept power series of sum function and its convergence radius, and the basic properties of the sum function of power series in convergence interval (refers to open interval) and convergence region; Solution of simple power series sum function: Fourier coefficient of elementary power series expansion function and Dlrichlei theorem of Fourier series: sine series and cosine series of Fourier series function on [-l, l].

Examination requirements

1. Understand the concepts of convergence and sum of convergent constant series, and master the basic properties of series and the necessary conditions for convergence.

2. Master the conditions of convergence and divergence of geometric series and P series.

3. Master the comparison of convergence of positive series and the ratio discrimination method, and use the root value discrimination method.

4. Master the Leibniz discriminant method of staggered series.

5. Understand the concepts of absolute convergence and conditional convergence of arbitrary series, and the relationship between absolute convergence and conditional convergence.

6. Understand the convergence domain of function term series and the concept of function.

7. Understand the concept of convergence radius of power series and master the solution of convergence radius, convergence interval and convergence domain of power series.

8. Knowing some basic properties of power series in its convergence interval (continuity of sum function, item-by-item differentiation, item-by-item integration), we will find the sum function of some power series in its convergence interval, and then find the sum of some series.

9. Understand the necessary and sufficient conditions for the function to expand into Taylor series.

10. Master maclaurin expansions of ex, sinx, cosx, ln( 1+x) and (1+x)α, and use them to indirectly expand some simple functions into power series.

1 1. Understand the concept of Fourier series and Dirichlet convergence theorem, expand the function defined on [- 1, L] into Fourier series, expand the function defined on [0, L] into sine series and cosine series, and write the expression of the sum of Fourier series.

Eight, ordinary differential equations

Examination content

The basic concept of ordinary differential equation: separable variable homogeneous differential equation, first-order linear differential equation, Bernoulli equation, full differential equation, some differential equations can be solved by simple variable substitution; The properties and structure theorems of solutions of reducible higher-order linear differential equations: the second-order homogeneous linear differential equation with constant coefficients is higher than the second-order homogeneous linear differential equation with constant coefficients; Simple application of second-order non-homogeneous linear differential equation with constant coefficients Euler differential equation

Examination requirements

1. Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.

2. Master the solutions of equations with separable variables and first-order linear equations.

3. Can solve homogeneous equations, Bernoulli equations and fully differential equations, and can replace some differential equations with simple variables.

4. The following equations will be solved by order reduction method: y (n) = f (x), y' = f (x, y'), y' = f (y, y').

5. Understand the properties of solutions of linear differential equations and the structure theorem of solutions.

6. Master the solutions of two groups of homogeneous linear differential equations with constant coefficients, and be able to solve some homogeneous linear differential equations with constant coefficients higher than the second order.

7. Polynomials, exponential functions, sine functions, cosine functions and their sum and product can be used to solve second-order non-homogeneous linear differential equations with constant coefficients.

8. Euler equation can be solved.

9. Can use differential equations to solve some simple application problems.

linear algebra

I. Determinants

Examination content

The concept and basic properties of determinant The expansion theorem of determinant by row (column)

Examination requirements

1. Understand the concept of determinant and master its properties.

2. The properties of determinant and determinant expansion theorem will be applied to calculate determinant.

Second, the matrix

Examination content

Concept of matrix, linear operation of matrix, multiplication of matrix, concept and properties of transposed inverse matrix of determinant matrix, necessary and sufficient condition of matrix reversibility, elementary transformation of adjoint matrix, rank matrix equivalent block matrix of elementary matrix and its operation.

Examination requirements

1. Understand the concepts and properties of matrix, identity matrix, quantitative matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric matrix.

2. Master the linear operation, multiplication, transposition and its operation rules of matrix, and understand the determinant properties of square matrix power and square matrix product.

3. Understand the concept of inverse matrix, master the properties of inverse matrix, the necessary and sufficient conditions of matrix reversibility, understand the concept of adjoint matrix, and use adjoint matrix to find inverse matrix.

4. Master the elementary transformation of matrix, understand the properties of elementary matrix and the concept of matrix equivalence, understand the concept of matrix rank, and master the method of finding matrix rank and inverse matrix by elementary transformation.

5. Understand the block matrix and its operation.

Third, the vector

Examination content

The linear combination of concept vectors of vectors and the linear representation of linear correlation of vector groups have nothing to do with the maximum linearity of linear independent vector groups. The orthogonal normalization method of vector space and related concepts between the rank of vector groups and the rank of matrix. N-dimensional vector space base transformation and coordinate transformation transformation matrix vector inner product linear independent vector groups orthogonal base orthogonal matrix specification and its properties.

Examination requirements

1. Understand the concepts of n-dimensional vectors, linear combinations of vectors and linear representations.

2. Understand the concepts of linear correlation and linear independence of vector groups, and master the related properties and discrimination methods of linear correlation and linear independence of vector groups.

3. Understand the concepts of maximal linearly independent group and rank of vector group, and find the maximal linearly independent group and rank of vector group.

4. Understand the concept of vector group equivalence and the relationship between the rank of matrix and the rank of its row (column) vector group.

5. Understand the concepts of N-dimensional star space, subspace, basement, dimension and coordinate.

6. Understand the formulas of base transformation and coordinate transformation, and find the transformation matrix.

7. Understand the concept of inner product and master the standard and standardized SChnddt method of linear independent vector group.

8. Understand the concepts of standard orthogonal basis and orthogonal matrix, and their properties.

Fourth, linear equations.

Examination content

Cramer's Law of Linear Equations Necessary and Sufficient Conditions for Homogeneous Linear Equations to Have Non-zero Solutions Necessary and Sufficient Conditions for Non-homogeneous Linear Equations to Have Solutions Properties and Structures of Solutions Basic Solution System of Homogeneous Linear Equations and General Solution of Non-homogeneous Linear Equations in General Solution Space.

Examination requirements

The length can be determined by Cramer's law.

2. Understand the necessary and sufficient conditions for homogeneous linear equations to have nonzero solutions and nonhomogeneous linear equations to have solutions.

3. Understand the concepts of basic solution system, general solution and solution space of homogeneous linear equations, and master the solution of basic solution system and general solution of homogeneous linear equations.

4. Understand the structure of solutions of nonhomogeneous linear equations and the concept of general solutions.

5. Master the method of solving linear equations with elementary line transformation.

Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix

Examination content

Concept and property similarity transformation of eigenvalues and eigenvectors of matrices, concept and necessary and sufficient conditions of similar diagonalization of property matrices, eigenvalues, eigenvectors and similar diagonal matrices of real symmetric matrices of similar diagonal matrices.

Examination requirements

1. Understand the concepts and properties of eigenvalues and eigenvectors of a matrix, and you will find the eigenvalues and eigenvectors of the matrix.

2. Understand the concept and properties of similar matrix and the necessary and sufficient conditions for matrix similarity diagonalization, and master the method of transforming matrix into similar diagonal matrix.

3. Master the properties of eigenvalues and eigenvectors of real symmetric matrices.

Sixth, the content of the second interview

Quadratic form and its matrix represent contract transformation and rank inertia theorem of quadratic form of contract matrix. The canonical form and canonical form of quadratic form are transformed into canonical quadratic form and the positive definiteness of its matrix by orthogonal transformation and matching method.

Examination requirements

1. Master quadratic form and its matrix representation, understand the concept of quadratic form rank, understand the concepts of contract change and contract matrix, and understand the concepts of standard form and standard form of quadratic form and inertia theorem.

2. Master the method of transforming quadratic form into standard form by orthogonal transformation, and can transform quadratic form into standard form by matching method.

3. Understand the concepts of positive definite quadratic form and positive definite matrix, and master their discrimination methods.

Probability and Mathematical Statistics

I. Random events and probabilities

Examination content

The relationship between random events and events in sample space and the operation of complete event group probability; Basic properties of conceptual probability; Classical probability, geometric probability, basic formula of conditional probability; Independent repeated testing of events.

Examination requirements

1. Understand the concept of sample space (basic event space), understand the concept of random events, and master the relationship and operation of events.

2. Understand the concepts of probability and conditional probability, master the basic properties of probability, calculate classical probability and geometric probability, and master the addition formula, subtraction formula, multiplication formula, total probability formula and Bayesian formula of probability.

3. Understand the concept of event independence and master the probability calculation with event independence; Understand the concept of independent repeated test and master the calculation method of related event probability.

Second, random variables and their probability distribution

Examination content

The concept and properties of random variables and their probability distribution, the probability distribution of discrete random variables, the probability density of continuous random variables, the probability distribution of common random variables, and the probability distribution of random variable functions.

Examination requirements

1. Understand the concept of random variables and their probabilistic market division. Understand the distribution function.

f(X)= P { X < = X }(-∞& lt； x & lt+∞)

Calculates the probability of events related to random variables.

2. Understand the concept and probability distribution of discrete random variables, and master 0-L distribution, binomial distribution, hypergeometric distribution, Poisson distribution and their applications.

3. Understand the conclusion and application conditions of Poisson theorem, and use Poisson distribution to approximately represent binomial distribution.

4. Understand the concept of continuous random variables and their probability density, and master uniform distribution, normal distribution N(μ, σ2), exponential distribution and their applications, where the parameter is λ (λ >; The density function of the exponential distribution of 0) is

5. Find the distribution of random variable function.

Three, two-dimensional random variables and their probability distribution

Examination content

Probability distribution, edge distribution and conditional distribution of multidimensional random variables and their distributions Probability density, marginal probability density and conditional density of two-dimensional continuous random variables Independence and correlation of random variables Probability distribution of two or more simple functions of random variables is commonly used.

Examination requirements

1. Understand the concept of multidimensional random variables, and understand the concept, properties and two basic forms of distribution of multidimensional random variables. Understand the probability distribution, edge distribution and conditional distribution of two-dimensional discrete random variables; Understand the probability density, edge density and conditional density of two-dimensional discrete random variables, and find the probability of related events of two-dimensional continuous random variables.

2. Understand the concepts of independence and irrelevance of random variables, and master the conditions of mutual independence of random variables.

3. Grasp the two-dimensional uniform distribution, understand the probability density of the two-dimensional normal distribution, and understand the probability meaning of the parameters.

4. Will find the distribution of simple functions of two random variables, and will find the distribution of simple functions of multiple independent random variables.

Fourth, the numerical characteristics of random variables

Check internal guests

Mathematical expectation (mean), variance and standard deviation of random variables and their properties Mathematical expectation moment, covariance correlation coefficient and their properties of random variable functions

Examination requirements

1. Understand the concept of digital characteristics of random variables (mathematical expectation, variance, standard deviation, covariance, correlation coefficient), and use the basic properties of digital characteristics to master the digital characteristics of common distribution.

2. According to the probability distribution of random variables, the mathematical expectation of their functions will be found.

Law of Large Numbers and Central Limit Theorem

Examination content

Chebyshev Inequality Chebyshev's Law of Large Numbers Bernoulli's Law of Large Numbers De Morville-… Les Theorem Levi-Onder Theorem

Examination requirements

1. Understanding Chebyshev Inequality.

2. Understand Chebyshev's law of large numbers, Bernoulli's law of large numbers and Sinchin's law of large numbers (the law of large numbers of independent and identically distributed random variable sequences).

3. Understand de moivre-Laplace Theorem (binomial distribution takes normal distribution as the limit distribution) and Levi-Lindbergh Theorem (central limit theorem of independent identically distributed random variable sequence).

Basic concepts of mathematical statistics of intransitive verbs

Examination content

Sample variance and sample moment x2 distribution T distribution F distribution Quantile normal population Some commonly used sampling distributions

Examination requirements

1. Understand the concepts of population, simple random sample, statistics, sample mean, sample variance and sample moment, where sample variance is defined as:

2. Understand the concepts and properties of x2 distribution, t distribution and f distribution, understand the concept of quantile and check it.

3. Understand some commonly used normal population sampling distributions.

Seven. parameter estimation

Examination content

Concept estimation of point estimation and estimated value Method of moment estimation Maximum likelihood estimation Method of estimation criterion Interval estimation Concept Interval estimation of mean and variance of a single normal population Interval estimation of mean difference and variance ratio of two normal populations.

Examination requirements

1. Understand the concepts of point estimation, estimator and parameter estimation.

2. Master moment estimation methods (first and second moments) and maximum likelihood estimation methods.

3. Understand the concepts of unbiased estimator, validity (minimum variance) and consistency (consistency), and verify unbiased estimator.

4。 In order to understand the concept of interval estimation, we will find the confidence interval of the mean and variance of a single normal population, and the confidence interval of the mean difference and variance ratio of two normal populations.

Eight hypothesis tests

Examination content

Two types of false hypothesis testing in significance testing Hypothesis testing of mean and variance of single and two normal populations.

Examination requirements

1. Understand the basic idea of significance test, master the basic steps of hypothesis test, and understand two possible errors in hypothesis test.

2. Master the hypothesis test of the mean and variance of single and two normal populations.

Test paper structure

(1) Test scores and test time: the full score of the test paper is 150, and the test time is 180 minutes.

(2) Content ratio: about 60% of higher education; Linear algebra is about 20%; Probability theory and mathematical statistics 20%;

(3) the proportion of questions: fill-in-the-blank questions and multiple-choice questions account for about 40%; Answer questions (including proof questions) about 60%